Hint

Answer

Their probability of meeting is 5/9

There are four events that can happen.
1. Bob arrives first and Alice meets him while he is waiting.
2. Bob arrives first, leaves and then Alice shows up.
3. Alice arrives first and Bob meets her while she is waiting.
4. Alice arrives first, leaves and then Bob shows up.

Also there are 3 main time segments that different things can happen during. from 12:00 12:00-12:20, 12:20-12:40, 12:40-1:00.

There is a 1/3 chance of Alice showing up in any of these times.

If Alice shows up from 12:20-12:40
Then there are 20 minutes before and after Alice's arrival that Bob could arrive and they would meet. So 40 minutes out of the hour that Bob could arrive and they will meet. (40/60)

If Alice shows up between 12:00 and 12:20 there are 20 minutes after her arrival, and all the time between 12:00 and when she arrived. This time averages out to be 10 minutes (remember we are assuming she showed up between 12:00 and 12:20) So that's 30 minutes out of the hour that Bob could show up and they would meet. (30/60)

From 12:40-1:00 you have a similar situation. There's twenty minutes before Alice's arrival that Bob could have shown up. And all the time till 1:00, which again averages out to 10 minutes. So again 30 minutes out of the hour.
(30/60)

Each of those three are equally likely with 1/3 probability so
1/3(30/60 + 40/60 + 30/60)